Contents

Summary and Analysis

Algebra II: Functions

Terms

Relations and Functions, page 2

page 1 of 2

Relations

A relation is a set of inputs and outputs, often written as ordered pairs
(input, output). We can also represent a relation as a mapping diagram or a
graph. For example, the relation can
be represented as:

Mapping Diagram of Relation

Lines connect the inputs with their outputs. The relation can also be represented as:

Graph of Relation

Functions

A function is a relation in which each input has only one output.

In the relation ,
y
is a function of
x
,
because for each input
x
(1, 2, 3, or 0), there is only one output
y
.
x
is not a function of
y
, because the input
y = 3
has multiple outputs:
x = 1
and
x = 2
.

Examples:

\:
y
is a function of
x
,
x
is a function
of
y
.

:
y
is not a function of
x
(
x = 3
has multiple outputs),
x
is a function of
y
.

:
y
is a function of
x
,
x
is not a
function of
y
(
y = 9
has multiple outputs).

:
y
is not a function of
x
(
x = 1
has multiple outputs),
x
is not a function of
y
(
y = 2
has multiple outputs).

The Line Test for Mapping Diagrams

To check if a relation is a function, given a mapping diagram of the relation,
use the following criterion: If each input has only one line connected to it,
then the outputs are a function of the inputs.

Example: In the following mapping diagram,
y
is a function of
x
, but
x
is not a function of
y
: